Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#101 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | C22⋊C4 |
| commutator | C22 | C2×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C22×C3 |
| socle | C2×C3 | C3×C22⋊C4 |
| prime radical | C22×C3 | C2×C3×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C2×C4×C9 ◁ G | |
57 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,4,3,5,7,6,8,9)(10,11,13,17,14,16,22,24)(12,18,19,25,20,15,21,23);(10,12,14,20)(11,15,16,18)(13,21,22,19)(17,25,24,23)99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 51);